Closure Properties of

2
.Homomorphisms
 A homomorphism on an alphabet is a function
that gives a string for each symbol in that
alphabet.
 Example: h(01010) = ababab.
 Example: h(0) = ab.
 Extend to strings by h(a1…an) = h(a1)…h(an). h(1) = ε.

Closure Under
Homomorphism
 If L is a regular language. and h is a
homomorphism on its alphabet.
a.
 Proof: Let E be a regular expression for L. then h(L)
= {h(w) | w is in L} is also a regular
language.Apply h to each symbol in E.
3
.
b.Language of resulting RE is h(L).

 h(R)  defines the language h(L)
 To prove: h(L) is regular. :
L(h(E)) = h(L(E))
.
 h(E)  replacing each symbol ∑ in E by h(a).e.
 Assumptions:
 E  RE on ∑. the
language of h(E) is same as the language we
get.Proof
 Given: A regular language. L on ∑ and a
homomorphism function. h. In other words. i. when we apply h to the language L(E).

Solution:
 Start with a DFA A for L
 Construct a DFA B for h-1(L) with:
• The same set of states. then h-1(L) is also a regular language.Closure Proof for Inverse
Homomorphism
To prove: If h is a homomorphism from alphabet
∑ to alphabet T.
11
.
• The same start state.
• Input alphabet = the symbols to which
homomorphism h applies.
• The same final states. and L is a regular language
over T.

Input a
h
Input
h(a) to A
Start
A
Accept/Reje
ct
• The transitions for B are computed by applying
h to an input symbol a and seeing where A would
go on sequence of input symbols h(a). δB(q. a) = δA(q. h(a))
.
Formally.